Mechanics & Industry
Volume 21, Number 3, 2020
|Number of page(s)||16|
|Published online||03 April 2020|
A multiobjective optimization of journal bearing with double parabolic profiles and groove textures under steady operating conditions
College of Power and Energy Engineering, Harbin Engineering University, Harbin, PR China
2 Chongqing Yuejin Machinery Plant Co., Ltd, Chongqing, PR China
* e-mail: email@example.com
Accepted: 20 February 2020
The double parabolic profiles can help journal bearing to reduce bushing edge wear, but it also reduces load carrying capacity and increases friction loss. To overcome these drawbacks, in this study, a multiobjective optimization of journal bearing with double parabolic profiles and groove textures is researched under steady operating conditions using Taguchi and grey relational analysis methods. Firstly, a lubrication model with journal misalignment, elastic deformation, asperity contact, thermal effect is established and formation cause of drawbacks is illustrated. Then, an orthogonal test with considering six factors, i.e., groove number, groove depth, groove length, axial width of double parabolic profiles, radial height of double parabolic profiles and groove location is conducted, meanwhile the effects and significances of each factor on response variables are revealed. Finally, an optimal parameters combination of six factors is determined by grey relational analysis, which gives maximum load carrying capacity and minimum friction loss. Overall, this study may give guidance on journal bearing design to enhance its tribological performance.
Key words: Double parabolic profiles / Groove textures / Load carrying capacity / Friction loss / Multiobjective optimization
© AFM, EDP Sciences 2020
Journal bearing is a critical component in practical engineering. As the journal misalignment, deformation, machining and installation errors are unavoidable, bushing edge wear is found in some applications, as illustrated in Figure 1. Earlier study  had shown the double parabolic profiles can reduce bushing edge wear, but it also reduces load carrying capacity and increases friction loss, which still need to be well solved.
For mechanical components, surface texturing has been widely used in the past decades to improve their contact performance . Specially, effects of surface textures on performance of journal bearing also attracted wide attentions to scholars. Ji et al.  employed sinusoidal waves to characterize rough surface, which showed the greater roughness ratio can suppress the hydrodynamic effect of textures significantly. Hence, it is necessary to minimize roughness of textured surface. Gu et al.  presented a mixed lubrication model to analyze the performance of groove textured journal bearing with non-Newtonian fluid operating from mixed to hydrodynamic lubrications. Their results showed the surface texturing can increase the asperity contacts, but the contact behaviors mainly arise in first cycle of start-up process. When journal bearing works under normal operating conditions, the wear due to asperity contacts will be small. Literatures [5–10] showed the partial textures can positively affect bearing performance, but the optimal locations are some different depending on geometrical parameters and working conditions . Yu  and Lin  showed the textures located at rising phase of pressure field increases load carrying capacity, while the textures located at falling phase of pressure field reduces load carrying capacity. However, Tala-Ighil et al. [8,9] showed the textures located in declining part of pressure field can generate extra hydrodynamic lift. Shinde and Pawar  pointed out among three partial grooving configurations (90–1800, 90–2700, 90–3600), the first configuration gives the maximum pressure increase while the last configuration gives the minimum frictional power loss. Their results indicate the optimal location also depends on the optimization target.
Studies mentioned above have shown the texture performance is affected by many factors, which makes it very complicated to obtain the optimal design. To address this issue, some optimization methods are adopted by researchers, including the genetic algorithm [11–13], neural network  and sequential quadratic programming , while these methods are somewhat difficult in mathematics. By contrast, Taguchi method may provide a handy way for this issue. Taguchi assumes that introducing quality concepts at design stage may be more valuable than inspection after manufacturing. Hence, Taguchi method aims to optimize process to minimize quality loss with objective functions such as “the-nominal-the-best”, “the-larger-the-better”, or “the-smaller-the-better” depending on experimental objectives . In practical engineering, this method uses the concept of orthogonal array to reduce the numbers of experiments, which facilitates to research multi-parameters concurrently and evaluate the effects of each parameter. Some studies [17–19] have already adopted Taguchi method to optimize surface textures for journal bearing to maximize load carrying capacity, minimize side leakage and friction loss.
Despite of remarkable progress on research of surface textures, few studies have been researched the journal bearing with double parabolic profiles and groove textures. The novelty of this study is to adopt the Taguchi and grey relational analysis methods to conduct a multiobjective optimization of journal bearing with double parabolic profiles and groove textures, i.e., double parabolic profiles are applied to eliminate bushing edge wear, while groove textures are applied to overcome the negative effects of the former, such as reduced load carrying capacity and increased friction loss. The results show this study may help journal bearing to enhance its tribological performance.
The bushing edge wear.
Figure 2 illustrates the layout of double parabolic profiles and groove textures investigated in this study. In Figure 2, B and db are the bushing width and thickness; Ly and Lz are the axial width and radial height of double parabolic profiles, whose equation can be described as δz = (Lz /Ly 2) y 2; (θs – θe ), wg, dg, lg and we, are the groove location, width, depth, length and gap, and their specific values will be introduced in Section 4.
Layout of double parabolic profiles and groove textures.
Figure 3 illustrates a misaligned journal bearing under external moment Me , whose lubricating oil is supplied through the axial oil feeding groove. For simplicity, only the misalignment in vertical plane yoz is considered.
As the elastic module of journal is much higher than that of bushing, only elastic deformation of bushing surface is considered. Thus, the nominal film thickness h is(1)where hg is the nominal film thickness without elastic deformation, which is(2)where c is the radial clearance, e the eccentricity of the midplane, ϕ the attitude angle of the midplane, y the axial coordinate, γ the misalignment angle, δz the clearance added by double parabolic profiles, δtex the clearance added by groove textures. Obviously, for journal bearing with plain profile, δz = δtex = 0, and for journal bearing with only double parabolic profiles, δtex = 0.
In this study, the elastic deformation δe is obtained by Winkler/Column model , which gives a simpler way than finite element method  to estimate elastic deformation and has been used in literatures [22–24]. The model assumes the local elastic deformation is only dependent on local film pressure, as expressed in equation (3) (3)where ν is the Poisson's ratio of alloy layer, E the elastic modulus of alloy layer, p the film pressure, d the thickness of alloy layer.
A misaligned journal bearing.
The Reynolds equation based on average flow model is utilized to determine the roughness effects on performance of journal bearing [25,26], as expressed in equation (4) (4) where μ is the viscosity of lubricating oil, p the film pressure, U1 and U2 the velocities of two surfaces, σ the standard deviation of combined roughness, φx , φy the pressure flow factors, φs the shear flow factor, hT the local film thickness.
For journal bearing under steady operating conditions, equation (4) can be expressed as followed by the variable transformation x = Rθ (5)where ω is angular velocity of journal.
The asperity contact model proposed by Greenwood and Tripp  is utilized here to estimate interaction effects of asperities, which is widely used in the analysis of rough surfaces contact of journal bearing. The asperity contact pressure Pasp is given by(6)where η is the number of asperities per unit area, β the mean radius of curvature of the asperities, σ the standard derivation, Ec the composite elastic modulus, F2.5 (h/σ) the Gaussian distribution function. Note the surface pattern parameter γ is assumed as 1, which means the roughness structures are isotropic.
It is assumed when journal bearing operates in mixed lubrication, the total friction force consists of hydrodynamic friction force arising from the shearing of lubricating oil and asperity contact friction force . Hence, the total friction force f is(7)where U = ω R, φf , φfs , φfp are shear stress factors, μasp boundary friction coefficient, here μasp = 0.02. The friction loss Pf can be calculated by(8)
The leakage flowrate Q1 from front-end plane of bearing and Q2 from rear end plane of bearings are (9)
As well known, the temperature of lubricating oil will increase and its viscosity will decrease during operations, so it is more accurate to adopt a variable viscosity model in calculation. In this study, an effective temperature is obtained based on the adiabatic flow hypothesis of lubricating oil, as shown below(11)where Te is the effective temperature of lubricating oil, Ti the inlet oil temperature, Pf the friction loss, Q the total leakage flowrate, ρ the density of lubricating oil, cl the specific heat of lubricating oil, k the correction factor and k = 0.9 . This simple method avoids the complex computation of thermohydrodynamic lubrication and has been used in the literatures .
CD40 lubricating oil is used here and its viscosity-temperature equation can be expressed as(12)where the unit of μ is Pa·s, and a = 6.163, b = 8.721 × 10−5,c = − 0.0455, respectively. Once the effective temperature is obtained, the effective viscosity can be calculated by the equation (12).
In this study, the external load is assumed as a pure moment whose direction is parallel to x axis, which only leads to a journal misalignment in vertical plane yoz. The static equilibrium of journal center can be described as(13)where Me is the external moment, Mt the resultant moment of hydrodynamic moment Moil and asperity contact moment Masp , namely Mt = Moil + Masp .
The load equilibrium equations along x and z axis are(14)where Mex and Mez are the external moment along x and z axis, Mtx and Mtz the resultant moment along x and z axis, which can be expressed as follows(15) where Moilx and Moilz are the hydrodynamic moment along x and z axis, Maspx and Maspz the asperity contact moment along x and z axis, which can be calculated by(16) (17)
Apply the finite difference method to discretize equation (5), then solve the difference equations by overrelaxation iterative method. Reynolds boundary conditions are used to determine the rupture zone of oil film, and the pressures in oil feeding groove and both bearing ends are assumed as zero. The discretized pressure can be calculated by(18)where pi,j (kp + 1) is the film pressure for node (i, j) at the (kp+1)th iteration, pi,j (kp ) the film pressure for node (i, j) at the kpth iteration, ωs the overrelaxation factor, here ωs =1.5. DDi,j, CSi,j, CNi,j, CEi,j, CWi,j, CCi,j are the difference coefficients during the pressure solution.
The film pressure convergence criteria at the kpth iteration is given by(19)where ϵp is the allowable precision of pressure solution, here ϵp = 10−5. nθ and ny are the nodes numbers along circumferential and axial direction.
Based on the equation (11), the effective temperature convergence criteria at the ktth iteration is given by(20)where ϵt is the allowable precision of effective temperature solution, here ϵt = 10−4.
When journal bearing operates in steady operations, it can be assumed the sum of hydrodynamic and asperity contact moments is approximately equal to the external moment, specifically, the motion of journal can be obtained by correcting eccentricity ratio ϵ (ϵ = e/c), attitude angle ϕ and misalignment angle γ. The correction strategies can be expressed as follows(21) (22) (23)where Me is the amplitude of Me , Mt is the amplitude of Mt , ωϕ , ωϵ , ωγ are the correction factors of ϕ, ϵ, γ, and ωϕ = 0.9, ωϵ = 10−2, ωγ = 10−5, respectively.
As mentioned above, the external load is a pure moment whose direction is parallel to the x axis. Accordingly, the equilibrium convergence criteria can be given by(24) (25)where errxz and errM are allowable precisions for the equilibrium calculation, here errxz = 10−3, errM = 10−3.
The whole computational process is shown in Figure 4, and the programing platform is Intel Core i7-4790 at 3.6GHz with 32GB RAM.
Before the following analysis, it is necessary to validate the model. The calculated pressures are compared with Ferron's experimental results , as illustrated in Figure 5. The comparisons show the calculated results agree well with the experimental ones, which confirms the validity of this model.
Flow chart of the computational process.
The detailed parameters of the journal bearing investigated in this study are listed in Table 1.
Mesh refinement analysis is conducted based on the journal bearing with plain profile. Various mesh schemes and corresponding minimum film thickness (hmin) are listed in Table 2. It can be seen that hmin is converged when the mesh is 1441 × 181. Considering the solving time and accuracy, 1441 × 181 mesh is adopted.
Here one typical case is conducted to prove the bushing edge wear can be reduced by double parabolic profiles. For journal bearing with double parabolic profiles, the chosen values of axial width Ly and radial height Lz are 10 mm and 6 μm, and the relative variation of each performance parameter is defined as(26)where δdpp X is the relative variation of parameter X, pp means plain profile and dpp means double parabolic profiles. Note in this study, the maximum film pressure (Pmax) is used to indirectly reflect the variations of load carrying capacity, smaller Pmax means better load carrying capacity and vice versa.
The comparison results are listed in Table 3. It can be observed that although the ϵ and γ both increase to some extent, the hmin has sharply increased and the Paspmax has reduced to zero in dpp case, which confirms its validity in term of reducing edge wear. Note ϕ has reduced a little, this variation trend indicates the bearing stability is also improved as the dpp reduces effective bearing width, which agrees with the common sense that using short bearing is favorable for stability improvement. However, the load carrying capacity is reduced as the dpp leads to a less capacity area. Hence, greater Pmax is generated and hydrodynamic friction force also increases which causes greater Pf. It is obvious that Q is increased as the dpp increases the film pressure gradient at bushing edge, so the comprehensive effects of greater Pf and Q lead to a tiny change of Te based on equation (11).
Figure 6 illustrates the variations of film thickness and pressure between pp and dpp case, in Figure 6a, Δh = hdpp − hpp , and in Figure 6b, ΔP = Pdpp − Ppp . It can be seen that, compared with pp case, the dpp case presents thicker oil film at bushing edge, which avoids the asperity contacts between journal and bushing surfaces, so the asperity contact pressure reduces to zero. However, in the region away from bushing edge, the oil film in dpp case is thinner than that in pp case, which causes greater film pressure here, i.e., the dpp can move the film pressure peak to inside region.
As mentioned above, although dpp can reduce bushing edge wear, it also reduces load carrying capacity and increases friction loss. The main motivation of this study is to overcome these drawbacks by groove textures, as illustrated in Figure 2. Considering the journal bearing with double parabolic profiles and groove textures (dppgt), six factors are involved in the analysis: groove number ng, depth dg, length lg, axial width of dpp Ly , radial height of dpp Lz , and location (θs – θe ). Three levels for each factor are listed in Table 4. Note the total area of groove textures, Sg = wg × lg × ng, is constant, while the groove width wg, gap we, and number ng are different. The values of wg and we for corresponding ng are listed in Table 5.
Clearly the study of six factors at three optional levels needs 3  simulation tests if full factorial designs are implemented. To reduce the computing cost, Taguchi method with orthogonal array L18 is adopted here, as shown in Table 6. L18 is commonly used and it is more concise than L27 array.
As we are mainly concerned with load carrying capacity and friction loss, only the results of Pmax and Pf are given in Table 7. Note reference group are the results of dpp case (Ly = 5, 10, 15 mm, Lz = 3, 6, 9 μm, totally 9 cases, each case will be repeated twice).
As can be seen from Table 7, compared with reference cases, the groove textures show positive effects only in No. 3, 4, 8, 11, 13 and 18 tests (shown in bold). Take the No. 8 test to explain this phenomenon: As illustrated in Figures 7 and 8, the groove textures can strengthen the hydrodynamic effect of lubricating oil and generate extra local pressure at the downstream of film pressure filed, which increases the load carrying capacity. Meanwhile, the friction force arising from the shearing of oil reduces with thicker oil film, which causes less friction loss. Note the groove locations of these six tests are all 200°–350°, which indicates the proper groove location is beneficial to performance enhancement.
While for the remaining tests, the groove textures show negative effects on either load carrying capacity or friction loss compared with reference cases. Take No. 5 and 15 tests to explain this phenomenon: As illustrated in Figures 9–12, the groove textures located at 180°–330° (No. 5 test) and 190°–340° (No. 15 test) increase film thickness at main loading region, so the continuous film pressure generation is destroyed and greater pressure is generated in multi-peaks at untextured land (region between two adjacent grooves). This phenomenon is more evident when the groove textures locates at 180°–330°. Meanwhile, the hydrodynamic friction loss also reduces as explained previously. Moreover, the friction loss in No. 1, 9, 12, and 16 orthogonal tests are greater than those in reference cases, coincidentally the groove locations of these four tests are all 180°–330°, which indicates the improper groove location may bring harmful effects on texture performance.
Based on the orthogonal tests, the main effect analysis and analysis of variance (ANOVA) are performed to show the effects and significance of each factor . The effects of six factors on load carrying capacity (Pmax) are illustrated in Figure 13. It is observed the optimal parameters combination is at groove number 120, depth 10 μm, length 50 mm, axial width of dpp 10 mm, radial height of dpp 3 μm, and location 200°–350°, which may give a best load carrying capacity. However, this combination does not exist in orthogonal table and another computation is conducted, denoted as No. 19 test. The results of this test are Pmax = 64.0170 MPa and Pf = 4105.1540 W, respectively, and Pmax of all 19 tests are illustrated in Figure 14. It can be seen the No. 8 test, i.e., groove number 30, depth 15 μm, length 60 mm, axial width of dpp 10 mm, radial height of dpp 3 μm, and location 200°–350°, gives a maximum load carrying capacity, with only 4.69% increases of Pmax than that in pp case.
Table 8 lists the results of ANOVA for load carrying capacity (Pmax). The columns represent the sources, degrees of freedom (DF), sum of squares (SS), mean of squares (MS), F-values, and F0.05 (2, 5). Table 8 shows the groove number ng, location θs – θe and depth dg are the significant factors at 95% confidence level as their F-values are greater than F0.05 (2, 5). The percentage contributions (PCR) of all factors are also given in Table 8, which shows the groove number ng is the most important factor whose percentage contribution is 38.66%, followed by location θs – θe and depth dg, whose percentage contribution are 25.78% and 16.85%.
The effects of six factors on the friction loss (Pf) are illustrated in Figure 15. It is clearly that Figure 15 suggests the optimal parameters combination is in the No. 6 test, i.e., groove number 60, depth 20 μm, length 60 mm, axial width of dpp 5 mm, radial height of dpp 3 μm, and location 190°–340°, which gives a minimum friction loss. The Pf of all 19 tests are illustrated in Figure 16, which also confirms the No. 6 test gives a minimum friction loss, 3643.9945 W, with 11.78% decrease than that in pp case.
Table 9 lists the results of ANOVA for friction loss (Pf), which shows the groove number ng, depth dg, axial width of dpp Ly, radial height of dpp Lz and location θs – θe are the significant factors at 95% confidence level. The percentage contributions of all factors are also given in Table 9, which shows the groove number ng is the most important factor whose percentage contribution is 33.45%, followed by the location θs – θe , depth dg, axial width of dpp Ly and radial height of dpp Lz whose percentage contributions are 21.57%, 21.00%, 11.39% and 8.04% respectively.
Note the interactions between factors are not considered in. In fact, the interactions can be neglected if the orthogonal table is reasonably designed. The ANOVA shows that, compared with the six factors, the PCR of errors are small, 6.62% of Pmax and 2.70% of Pf, which indicates the interactions hidden in error are very limited. It can be found the interactions are also neglected in literatures [17–19].
From the above analysis, it can be observed the No. 8 test gives a maximum load carrying capacity while the No. 6 test gives a minimum friction loss. To find an optimal configuration, the grey relational analysis (GRA) method is used for this multiobjective optimization . The main steps of GRA are listed as follows:
Step 1: Solution for normalized sequence.
As the optimization objectives are increasing load carrying capacity (smaller Pmax) and reducing friction loss (smaller Pf), the “the-smaller-the-better” criterion is used to normalize the orthogonal test results between 0 and 1, as shown below(27)where the Xi * (k) is the normalized sequence, xi(k) the sequence of Pmax or Pf, k =1, 2 (1 for Pmax and 2 for Pf), and i = 1, 2,…, 19 (test No.).
Step 2: Solution for deviation sequence.
Step 3: Solution for grey relational coefficient (GRC).
The GRC is calculated depending on the deviation sequence to describe the correlation between the reference and normalized sequences, as shown below(29)where Δ min and Δ max are the minimum and maximum values of Δ 0i (k), ζ the distinguishing factor between 0 and 1, here ζ = 0.5, which means the increasing load carrying capacity and reducing friction loss are equally important.
Step 4: Solution for grey relational grade (GRG).
The evaluation criteria for a multiobjective optimization is based on GRG, which is the mean of the GRCs, as shown in equation (30). The larger GRG means the parameter configuration approaches the optimal one.(30)Based on above four steps, the GRGs of all 19 tests are calculated and illustrated in Figure 17. It is observed the GRG of No. 8 test is the largest one, which means when both considering the load carrying capacity and friction loss, the parameters of No. 8 test, i.e., groove number 30, depth 15 μm, length 60 mm, axial width of dpp 10 mm, radial height of dpp 3 μm, and location 200°–350°, is the optimal parameters combination. In this case, the Pmax is 62.8141 MPa and Pf is 4113.0399 W, with only 4.69% increase while 0.42% decrease than those in the pp case, and meanwhile the bushing edge wear is totally eliminated.
At last, it should be noted that the Taguchi and grey relational analysis methods gives the optimal parameters combination only limited to the preassigned parameter levels, which may miss the really optimal design. However, they are relatively simple and practicable, which facilitates the research of multi-parameters and evaluation of parameter effects.
Detailed parameters of the journal bearing.
Mesh schemes for mesh refinement analysis.
Comparisons between journal bearing with plain profile (pp) and double parabolic profiles (dpp).
(a) Differences Δh, (b) Differences ΔP between pp and dpp cases.
Six control factors and their optional levels.
Groove sizes for their optional levels.
Orthogonal array L18.
Results of orthogonal tests and reference group.
(a) Film thickness of No. 8 test, (b) Film thickness of reference case 8.
(a) Film pressure of No. 8 test, (b) Film pressure of reference case 8.
(a) Film thickness of No. 5 test, (b) Film thickness of reference case 5.
(a) Film pressure of No. 5 test, (b) Film pressure of reference case 5.
(a) Film thickness of No. 15 test, (b) Film thickness of reference case 15.
(a) Film pressure of No. 15 test, (b) Film pressure of reference case 15.
Main effect plots of Pmax (MPa).
Pmax of all 19 tests.
ANOVA for Pmax.
Main effect plots of Pf (W).
Pf of all 19 tests.
ANOVA for Pf.
GRGs of all 19 tests.
In this study, based on Taguchi and GRA methods, a multiobjective optimization of journal bearing with double parabolic profiles and groove textures is researched under steady operating conditions. Following conclusions can be drawn from numerical results:
Compared with plain profile, double parabolic profiles can eliminate bushing edge wear, and the stability of journal bearing can be also improved. However, it also has visible drawbacks, such as the reduced load carrying capacity (with 13.51% increase of Pmax) and increased friction loss (with 3.46% increase of Pf).
Compared with double parabolic profiles, double parabolic profiles with proper groove textures can increase the load carrying capacity due to extra local pressure. Meanwhile, the friction loss also reduces with thicker oil film. The improper groove textures can deteriorate the performance as they will destroy continuous film pressure generation and increase friction loss.
The main effects analysis based on orthogonal test results shows that, for load carrying capacity, the No.8 test gives a minimum Pmax, 62.8481 MPa, with only 4.69% increase than that in the case of plain profile. While for friction loss, the No. 6 test gives a minimum Pf, 3643.9945 W, with 11.78% decrease than that in the case of plain profile.
The ANOVA shows that, for load carrying capacity, the groove number ng and groove location θs – θe are the significant factors at 95% confidence level, whose percentage contributions are 38.66%, 25.78%, 16.85% respectively. While for friction loss, the groove number ng, groove location θs – θe , groove depth dg, axial width of double parabolic profiles Ly and radial height of double parabolic profiles Lz are the significant factors at 95% confidence level, whose percentage contributions are 33.45%, 21.57%, 21.00%, 11.39%, and 8.04% respectively.
The GRA shows that, the parameters combination of No. 8 test, i.e., groove number 30, groove depth 15 μm, groove length 60 mm, axial width of double parabolic profiles 10 mm, radial height of double parabolic profiles 3 μm, and groove location 200°–350°, is the optimal solution: Pmax is 62.8141 MPa and Pf is 4113.0399 W, with only 4.69% increase while 0.42% decrease than those in the case of plain profile, meanwhile the bushing edge wear is eliminated.
The cavitation area of oil film determined by Reynolds boundary conditions may be less accurate than mass-conservative treatment, which is a limitation of this study. In future work, cavitation effects will be treated in mass-conserving way, and other optimization methods will also be tried for journal bearing to find the optimal texture design.
This work is supported by the National Natural Science Foundation of China (51809057) and Fundamental Research Funds for the Central Universities (3072019CFM0302).
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Cite this article as: C. Liu, N. Zhong, X. Lu, B. Zhao, A multiobjective optimization of journal bearing with double parabolic profiles and groove textures under steady operating conditions, Mechanics & Industry 21, 305 (2020)
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